Resampling Methods

DDD: Elements of Statistical Machine Learning & Politics of Data

Ayush Patel

At Azim Premji University, Bhopal

29 Jan, 2026

Hello

I am Ayush.

I am a researcher working at the intersection of data, development and economics.

I am a RStudio (Posit) certified tidyverse Instructor.

I am a Researcher at Oxford Poverty and Human development Initiative (OPHI), at the University of Oxford.

Did you come prepared?

  • You have installed R. If not see this link.

  • You have installed RStudio/Positron/VScode or any other IDE. It is recommended that you work through an IDE

  • You have the libraries {tidyverse} {caret} {ISLR} {ISLR2} {openintro} {broom} {MASS} {nnet} {e1071} {tidymodels} installed

Learning Goals

  1. Apply resampling methods for Model Assessment (performance)
  2. Apply resampling methods for Model Selection (flexiblity)

Resampling Methods

The process involves taking several samples from the training data and fitting the model of interest to learn more about the model.

Motivation

Training error rates are not the best to gauge the model’s performance

Test data is not usually available

But test error rate are better to assess a model’s performance

One of the ways to handle this is by using resampling methods to estimate tert error rates

Validation Set Approach

Split the observation randomly into training and Validation set.

Use the training set to build/train the model.

Use the model to predict the responses of the validation set.

Compare various quality of fit and performance stats using validation set

Data

Auto as an example to learn Cross Validation

Model in question

lm(mpg ~ horsepower, Auto) |>
    broom::glance()
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC deviance df.residual nobs
0.6059483 0.6049379 4.905757 599.7177 0 1 -1178.662 2363.324 2375.237 9385.916 390 392
lm(mpg ~ poly(horsepower,2), Auto) |>
    broom::glance()
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC deviance df.residual nobs
0.687559 0.6859527 4.373921 428.0176 0 2 -1133.177 2274.354 2290.239 7442.029 389 392

Model in question

 lm(mpg ~ horsepower, Auto) |>
    broom::augment() |> 
    summarise(mse = mean(.resid ^2))
# A tibble: 1 × 1
    mse
  <dbl>
1  23.9
lm(mpg ~ poly(horsepower,2), Auto) |>
    broom::augment() |> 
    summarise(mse = mean(.resid ^2))
# A tibble: 1 × 1
    mse
  <dbl>
1  19.0

Model in question

Selection: Would models with more flexibility prove better?

Performance: Does the second model have better MSE? Will this be true for any sample?

Validatoin Set Approach

Split sample in training and Validation set

rsample::initial_split(Auto,prop = .5) -> Auto_split 

Auto_split
<Training/Testing/Total>
<196/196/392>
Auto_train <- rsample::training(Auto_split)

Auto_validation_set <- rsample::testing(Auto_split)

validation Set Approach

Train model using training data

lm(mpg ~  poly(horsepower,3), Auto_train) -> trained_model

head(predict(trained_model, Auto_validation_set))
       1        2        3        4        5        6 
13.69500 14.79623 15.80218 12.64127 12.75884 13.42536 
tibble(
    y = Auto_validation_set$mpg,
    y_hat = predict(trained_model, Auto_validation_set),
    resid_sqr = (y - y_hat)^2
) |>
    summarise(mse = mean(resid_sqr, na.rm = T))
# A tibble: 1 × 1
    mse
  <dbl>
1  20.8

Validation Set Approach

Get MSE for all powers from 2 to 10

# define a funciton

get_mse <- function(pow){

    # train the model - using trainging data
    lm(mpg ~ poly(horsepower,pow), Auto_train) -> trained_model

    # get mse
    tibble(
        y = Auto_validation_set$mpg,
        y_hat = predict(trained_model, Auto_validation_set),
        resid_sqr = (y - y_hat)^2
    ) |>
        summarise(mse = mean(resid_sqr, na.rm = T)) |>
        pull(mse) -> mse

    # Store appropriately

    tibble(
        pow_flexibility = pow,
        mse = mse
    )

}

Validation set Approach

Get MSE for all powers of 1 to 10

map(
    c(1:10),
    get_mse
) |>
    list_rbind() |>
    ggplot(aes(pow_flexibility,mse))+
    geom_point()+
    geom_line()+
    labs(
    x = "Flexibility",
    y = "MSE"
    )+
    theme_minimal()

Validation set Approach

What happens if the sample changes

For each power (level of flexibility) I run the model 10 times.

Validation Set Approach

adjust funciton to split every time when executed

get_mse_spread <- function(pow){

    # repeats split for every iteration

    rsample::initial_split(Auto,prop = .5) -> Auto_split 

    Auto_train <- rsample::training(Auto_split)

    Auto_validation_set <- rsample::testing(Auto_split)

    # train the model - using trainging data
    lm(mpg ~ poly(horsepower,pow), Auto_train) -> trained_model

    # get mse
    tibble(
        y = Auto_validation_set$mpg,
        y_hat = predict(trained_model, Auto_validation_set),
        resid_sqr = (y - y_hat)^2
    ) |>
        summarise(mse = mean(resid_sqr, na.rm = T)) |>
        pull(mse) -> mse

    # Store appropriately

    tibble(
        pow_flexibility = pow,
        mse = mse
    )

    
}

Validation Set Approach

Execute 10 times for every power to get mse spread

map(
    rep(c(1:10),10) |> sort(),
    get_mse_spread
) |>
    list_rbind() |> 
    mutate(
        iteration = rep(letters[1:10],10)
    ) |>
    ggplot(aes(pow_flexibility,mse, colour = iteration))+
    geom_point()+
    geom_line(aes(group = iteration))+
    labs(
    x = "Flexibility",
    y = "MSE"
    )+
    theme_minimal() +
    theme(
        legend.position = "none"
    )

Validation Set Approach

Validation Set Approach

Potential Problems

Validation set estimate of test MSE can be highly variable

Does not do well when number of observation are fewer. We end up not using all informaiton that we have to train the model. It might overestimate the test error rate.

Leave-one-out Cross-Validation

Addresses the issue with Validation Set Approach, but is computationally costly.

For a data set of n observations, the model is fit n times using.

The model is fit using n-1 observations as training data in a manner where each observation is left out of the training data exactly once.

Test MSE is estimated using the error rates of all the left out observation predictions.

Leave-one-out Cross-Validation

Begin by creating the splits

rsample::loo_cv(Auto) -> Auto_loocv_split

Auto_loocv_split
# Leave-one-out cross-validation 
# A tibble: 392 × 2
   splits          id        
   <list>          <chr>     
 1 <split [391/1]> Resample1 
 2 <split [391/1]> Resample2 
 3 <split [391/1]> Resample3 
 4 <split [391/1]> Resample4 
 5 <split [391/1]> Resample5 
 6 <split [391/1]> Resample6 
 7 <split [391/1]> Resample7 
 8 <split [391/1]> Resample8 
 9 <split [391/1]> Resample9 
10 <split [391/1]> Resample10
# ℹ 382 more rows

Leave-one-out Cross-Validation

Function to Fit model on every training split and get error for each test observation

get_error_obs <- function(split,pow){

    Auto_loocv_train <- rsample::training(Auto_loocv_split$splits[[split]])

    Auto_loocv_test <- rsample::testing(Auto_loocv_split$splits[[split]])

    mod <- lm(mpg ~ poly(horsepower,pow),data = Auto_loocv_train)

    broom::augment(mod, newdata = Auto_loocv_test) |>
        select(
            mpg, `.fitted`,`.resid`
        ) |>
            mutate(
                split = split,
                pow = pow
            )

}

Leave-one-out Cross-Validation

Execute Function to get errors for all test obs.

pmap(
    list(
        split = rep(c(1:391),10),
        pow = rep(c(1:10),391) |> sort()
    ),
    get_error_obs
) |>
    list_rbind() -> error_obs_tab

error_obs_tab |>
    kableExtra::kable() |>
    kableExtra::kable_styling() |>
    kableExtra::scroll_box(width = '100%', height = '500px')

Leave-one-out Cross-Validation

mpg .fitted .resid split pow
25.8 25.413055 0.3869454 1 1
27.0 26.042634 0.9573660 2 1
23.8 26.527802 -2.7278023 3 1
24.2 20.984953 3.2150473 4 1
13.0 15.011474 -2.0114744 5 1
39.0 29.784234 9.2157662 6 1
15.0 16.266917 -1.2669167 7 1
44.0 31.637647 12.3623526 8 1
18.5 16.245331 2.2546692 9 1
10.0 5.902373 4.0976267 10 1
16.0 13.073093 2.9269072 11 1
25.0 28.745683 -3.7456834 12 1
25.0 27.316586 -2.3165864 13 1
31.0 29.352066 1.6479339 14 1
37.3 29.005401 8.2945990 15 1
18.0 19.421271 -1.4212706 16 1
34.5 28.860778 5.6392222 17 1
24.5 26.050205 -1.5502049 18 1
28.0 27.305794 0.6942061 19 1
34.0 28.863090 5.1369105 20 1
20.2 26.063227 -5.8632270 21 1
39.4 28.838124 10.5618764 22 1
15.0 16.266917 -1.2669167 23 1
28.0 28.097903 -0.0979025 24 1
23.0 24.629235 -1.6292350 25 1
26.6 26.677156 -0.0771561 26 1
13.0 17.070300 -4.0703004 27 1
17.0 16.254582 0.7454181 28 1
18.0 27.660299 -9.6602991 29 1
31.0 31.733290 -0.7332904 30 1
13.0 16.279252 -3.2792515 31 1
16.5 18.160748 -1.6607476 32 1
16.0 3.252290 12.7477103 33 1
19.4 25.748324 -6.3483241 34 1
28.4 25.722036 2.6779644 35 1
16.5 21.007870 -4.5078696 36 1
23.0 26.847625 -3.8476250 37 1
16.0 11.467612 4.5323881 38 1
26.0 25.412489 0.5875109 39 1
24.5 30.500971 -6.0009710 40 1
13.0 19.597574 -6.5975743 41 1
13.0 19.439719 -6.4397194 42 1
15.0 19.432340 -4.4323399 43 1
27.0 25.726125 1.2738751 44 1
33.0 31.559830 1.4401696 45 1
16.0 16.260749 -0.2607493 46 1
10.0 8.336464 1.6635364 47 1
24.0 25.418144 -1.4181439 48 1
27.4 27.307952 0.0920476 49 1
26.0 22.088938 3.9110624 50 1
33.5 26.812367 6.6876334 51 1
24.0 25.734888 -1.7348878 52 1
14.0 13.890508 0.1094918 53 1
18.0 24.959442 -6.9594423 54 1
16.0 16.260749 -0.2607493 55 1
13.0 19.439719 -6.4397194 56 1
11.0 16.291586 -5.2915862 57 1
30.9 28.086109 2.8138907 58 1
17.0 19.424960 -2.4249603 59 1
19.0 24.164741 -5.1647411 60 1
26.8 21.769922 5.0300781 61 1
14.0 16.589919 -2.5899192 62 1
20.2 17.985235 2.2147655 63 1
30.0 26.665987 3.3340132 64 1
14.0 14.686106 -0.6861062 65 1
29.0 28.886206 0.1137940 66 1
15.0 28.630619 -13.6306187 67 1
30.5 29.988821 0.5111787 68 1
26.0 25.254229 0.7457710 69 1
12.0 14.701979 -2.7019794 70 1
30.7 27.928714 2.7712857 71 1
12.0 8.622992 3.3770076 72 1
18.2 18.628605 -0.4286050 73 1
14.0 18.330089 -4.3300888 74 1
33.8 29.338068 4.4619324 75 1
17.0 22.587489 -5.5874892 76 1
17.6 19.581002 -1.9810018 77 1
25.4 27.791031 -2.3910306 78 1
30.5 27.613116 2.8868842 79 1
15.0 16.266917 -1.2669167 80 1
13.0 15.487235 -2.4872353 81 1
32.9 24.128709 8.7712905 82 1
20.2 25.745987 -5.5459873 83 1
30.0 29.357066 0.6429344 84 1
16.5 11.461335 5.0386654 85 1
31.6 28.241400 3.3585996 86 1
24.3 25.734012 -1.4340115 87 1
19.0 26.543238 -7.5432375 88 1
11.0 11.530385 -0.5303853 89 1
22.5 25.739269 -3.2392692 90 1
22.0 23.365648 -1.3656485 91 1
14.0 5.805522 8.1944784 92 1
20.8 26.537449 -5.7374493 93 1
19.0 24.956729 -5.9567291 94 1
13.0 13.103286 -0.1032855 95 1
20.5 24.160853 -3.6608528 96 1
27.0 26.359202 0.6407981 97 1
13.0 13.899464 -0.8994636 98 1
24.0 24.784866 -0.7848660 99 1
13.0 16.279252 -3.2792515 100 1
15.0 9.867453 5.1325468 101 1
17.5 24.960799 -7.4607989 102 1
21.5 22.575741 -1.0757414 103 1
29.5 29.200970 0.2990302 104 1
16.0 22.590100 -6.5900998 105 1
31.8 29.664764 2.1352357 106 1
26.0 27.471529 -1.4715288 107 1
15.0 17.059469 -2.0594686 108 1
21.0 28.604278 -7.6042780 109 1
38.0 29.317070 8.6829301 110 1
18.0 26.069889 -8.0698895 111 1
20.0 25.746572 -5.7465715 112 1
13.0 13.103286 -0.1032855 113 1
23.0 26.054748 -3.0547475 114 1
38.0 26.482140 11.5178602 115 1
16.9 15.459906 1.4400938 116 1
31.5 28.716401 2.7835988 117 1
35.0 26.018407 8.9815932 118 1
41.5 27.885883 13.6141165 119 1
25.1 26.048388 -0.9483879 120 1
13.0 16.279252 -3.2792515 121 1
26.0 28.106036 -2.1060358 122 1
18.6 22.583312 -3.9833122 123 1
17.0 17.841579 -0.8415792 124 1
21.0 26.219434 -5.2194338 125 1
20.6 23.369230 -2.7692297 126 1
37.0 29.637372 7.3626284 127 1
22.0 22.257934 -0.2579341 128 1
25.5 25.888832 -0.3888323 129 1
14.0 16.273084 -2.2730841 130 1
35.0 29.016315 5.9836851 131 1
32.0 29.347067 2.6529334 132 1
18.0 23.375881 -5.3758806 133 1
22.0 28.599888 -6.5998879 134 1
31.0 26.504649 4.4953505 135 1
31.9 28.714599 3.1854008 136 1
39.1 30.728303 8.3716966 137 1
25.0 22.566604 2.4333958 138 1
29.0 31.748005 -2.7480047 139 1
32.8 31.720047 1.0799525 140 1
22.4 22.573392 -0.1733918 141 1
15.0 24.175110 -9.1751100 142 1
28.0 25.723204 2.2767960 143 1
15.0 16.266917 -1.2669167 144 1
16.0 17.846332 -1.8463320 145 1
36.1 30.431365 5.6686347 146 1
27.0 26.042634 0.9573660 147 1
26.6 23.353881 3.2461185 148 1
37.2 29.636318 7.5636820 149 1
14.0 4.979161 9.0208390 150 1
20.0 25.587995 -5.5879954 151 1
19.2 23.372811 -4.1728110 152 1
36.0 28.223048 7.7769519 153 1
34.7 29.965510 4.7344899 154 1
15.5 17.532034 -2.0320335 155 1
9.0 9.479539 -0.4795388 156 1
25.5 24.780843 0.7191570 157 1
12.0 11.517831 0.4821694 158 1
27.9 23.350556 4.5494439 159 1
31.3 28.084483 3.2155174 160 1
17.0 20.215813 -3.2158128 161 1
18.1 21.003108 -2.9031077 162 1
15.0 17.059469 -2.0594686 163 1
26.0 27.312989 -1.3129889 164 1
19.0 22.898659 -3.8986593 165 1
27.2 28.735772 -1.5357725 166 1
20.0 20.680902 -0.6809018 167 1
23.0 27.641426 -4.6414258 168 1
29.0 28.093836 0.9061641 169 1
19.8 26.540665 -6.7406650 170 1
18.0 23.375881 -5.3758806 171 1
18.5 22.583573 -4.0835733 172 1
32.0 24.763410 7.2365901 173 1
21.6 21.784222 -0.1842219 174 1
32.2 28.080823 4.1191773 175 1
36.0 28.853843 7.1461572 176 1
22.3 26.056867 -3.7568674 177 1
23.0 24.945877 -1.9458765 178 1
21.0 26.378100 -5.3781005 179 1
14.0 16.273084 -2.2730841 180 1
19.0 24.164741 -5.1647411 181 1
34.1 29.652648 4.4473517 182 1
17.5 22.586184 -5.0861839 183 1
18.0 24.642507 -6.6425068 184 1
14.0 16.273084 -2.2730841 185 1
17.5 17.839203 -0.3392029 186 1
19.4 17.830173 1.5698274 187 1
24.0 24.626581 -0.6265807 188 1
24.0 22.569215 1.4307852 189 1
15.0 16.266917 -1.2669167 190 1
36.1 29.484331 6.6156688 191 1
26.0 28.106036 -2.1060358 192 1
21.0 23.051731 -2.0517312 193 1
16.0 23.380997 -7.3809967 194 1
31.5 29.191229 2.3087713 195 1
17.0 22.587489 -5.5874892 196 1
16.0 24.172518 -8.1725178 197 1
27.0 25.726125 1.2738751 198 1
28.1 27.305434 0.7945659 199 1
16.0 23.380997 -7.3809967 200 1
22.0 27.963217 -5.9632169 201 1
16.0 16.419502 -0.4195019 202 1
15.0 13.083157 1.9168430 203 1
20.0 24.162149 -4.1621489 204 1
24.0 24.943163 -0.9431634 205 1
29.0 32.226840 -3.2268398 206 1
31.0 26.978830 4.0211698 207 1
20.0 26.063833 -6.0638327 208 1
25.0 28.110102 -3.1101024 209 1
43.1 32.272088 10.8279120 210 1
25.0 27.157995 -2.1579946 211 1
26.0 32.731904 -6.7319045 212 1
36.4 29.325069 7.0749310 213 1
23.0 26.371801 -3.3718010 214 1
19.2 17.036722 2.1632784 215 1
18.1 17.994959 0.1050411 216 1
33.7 28.074723 5.6252773 217 1
17.5 17.045929 0.4540713 218 1
28.0 27.464160 0.5358399 219 1
24.0 28.114169 -4.1141691 220 1
18.0 26.069889 -8.0698895 221 1
31.0 29.193664 1.8063361 222 1
20.6 22.578091 -1.9780909 223 1
26.4 26.044451 0.3555490 224 1
12.0 4.205438 7.7945615 225 1
15.0 22.592710 -7.5927104 226 1
17.7 13.857373 3.8426267 227 1
19.1 25.749200 -6.6492004 228 1
22.0 24.156965 -2.1569645 229 1
23.7 24.152558 -0.4525578 230 1
20.5 23.369485 -2.8694855 231 1
28.0 25.406834 2.5931657 232 1
32.4 28.554231 3.8457694 233 1
13.0 12.305295 0.6947052 234 1
14.0 12.294031 1.7059689 235 1
32.0 26.659417 5.3405833 236 1
13.0 5.829734 7.1702655 237 1
29.8 25.717946 4.0820537 238 1
43.4 32.269652 11.1303483 239 1
16.0 24.172518 -8.1725178 240 1
21.5 22.575741 -1.0757414 241 1
30.0 29.198535 0.8014655 242 1
37.7 30.106474 7.5935264 243 1
28.0 25.723204 2.2767960 244 1
14.5 15.952834 -1.4528336 245 1
18.0 24.642507 -6.6425068 246 1
27.5 24.933667 2.5663327 247 1
22.0 26.374951 -4.3749507 248 1
36.0 26.015378 9.9846216 249 1
33.0 28.235561 4.7644390 250 1
32.3 29.345567 2.9544332 251 1
29.0 28.886206 0.1137940 252 1
37.0 29.164441 7.8355594 253 1
34.0 26.021435 7.9785648 254 1
15.5 9.859746 5.6402535 255 1
22.0 24.473566 -2.4735660 256 1
44.6 29.284074 15.3159265 257 1
23.0 24.945877 -1.9458765 258 1
15.5 17.056761 -1.5567606 259 1
16.2 18.953404 -2.7534044 260 1
15.0 28.630619 -13.6306187 261 1
18.0 24.167333 -6.1673333 262 1
25.0 24.940450 0.0595498 263 1
18.0 24.167333 -6.1673333 264 1
32.0 29.663711 2.3362893 265 1
19.2 20.208576 -1.0085761 266 1
20.0 23.845549 -3.8455488 267 1
20.0 24.162149 -4.1621489 268 1
26.0 29.377063 -3.3770634 269 1
37.0 25.381388 11.6186123 270 1
15.5 13.078125 2.4218751 271 1
15.0 8.569085 6.4309155 272 1
19.0 25.749492 -6.7494925 273 1
26.0 28.900076 -2.9000759 274 1
46.6 29.586801 17.0131994 275 1
14.0 17.855837 -3.8558375 276 1
25.0 25.415316 -0.4153165 277 1
27.0 26.834193 0.1658068 278 1
29.9 29.674773 0.2252269 279 1
29.5 28.725411 0.7745889 280 1
14.0 15.797510 -1.7975104 281 1
36.0 30.747890 5.2521098 282 1
22.0 24.948590 -2.9485897 283 1
28.0 26.039606 1.9603944 284 1
32.0 28.872336 3.1276639 285 1
25.0 21.774872 3.2251281 286 1
34.3 27.598772 6.7012280 287 1
28.0 26.356052 1.6439478 288 1
12.0 13.590584 -1.5905842 289 1
19.0 24.639852 -5.6398524 290 1
19.0 22.265902 -3.2659020 291 1
14.0 16.273084 -2.2730841 292 1
14.0 16.273084 -2.2730841 293 1
34.1 29.178565 4.9214348 294 1
15.5 21.010846 -5.5108458 295 1
20.0 22.579657 -2.5796573 296 1
26.5 28.580132 -2.0801323 297 1
30.0 29.357066 0.6429344 298 1
18.0 23.375881 -5.3758806 299 1
25.4 21.615349 3.7846514 300 1
26.0 27.630102 -1.6301018 301 1
23.9 24.626846 -0.7268461 302 1
34.4 29.651068 4.7489321 303 1
14.0 16.273084 -2.2730841 304 1
16.0 16.260749 -0.2607493 305 1
24.0 28.114169 -4.1141691 306 1
11.0 6.694717 4.3052834 307 1
32.4 28.080009 4.3199907 308 1
28.8 21.764422 7.0355780 309 1
35.1 30.437366 4.6626342 310 1
13.0 12.305295 0.6947052 311 1
17.6 26.547739 -8.9477394 312 1
18.0 24.167333 -6.1673333 313 1
13.0 16.279252 -3.2792515 314 1
20.0 24.954016 -4.9540160 315 1
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18.0 24.692809 -6.6928094 111 10
20.0 24.051216 -4.0512160 112 10
13.0 13.634979 -0.6349785 113 10
23.0 24.647106 -1.6471059 114 10
38.0 25.521851 12.4781493 115 10
16.9 14.157400 2.7426003 116 10
31.5 31.093319 0.4066806 117 10
35.0 24.537417 10.4625827 118 10
41.5 28.975656 12.5243438 119 10
25.1 24.627910 0.4720896 120 10
13.0 14.903582 -1.9035823 121 10
26.0 29.548565 -3.5485654 122 10
18.6 20.381408 -1.7814081 123 10
17.0 16.511918 0.4880824 124 10
21.0 24.994857 -3.9948573 125 10
20.6 20.925638 -0.3256380 126 10
37.0 33.161522 3.8384775 127 10
22.0 20.123196 1.8768038 128 10
25.5 24.307512 1.1924880 129 10
14.0 14.882748 -0.8827481 130 10
35.0 31.813806 3.1861938 131 10
32.0 32.576844 -0.5768445 132 10
18.0 20.960275 -2.9602745 133 10
22.0 30.809330 -8.8093298 134 10
31.0 25.589935 5.4100651 135 10
31.9 31.088515 0.8114852 136 10
39.1 34.630222 4.4697783 137 10
25.0 20.276401 4.7235993 138 10
29.0 35.394745 -6.3947448 139 10
32.8 35.054226 -2.2542262 140 10
22.4 20.319060 2.0809400 141 10
15.0 21.744933 -6.7449330 142 10
28.0 23.980992 4.0190079 143 10
15.0 14.861914 0.1380861 144 10
16.0 16.535702 -0.5357025 145 10
36.1 34.457802 1.6421975 146 10
27.0 24.610543 2.3894570 147 10
26.6 20.845707 5.7542925 148 10
37.2 33.158143 4.0418572 149 10
14.0 12.489493 1.5105071 150 10
20.0 23.758327 -3.7583271 151 10
19.2 20.944288 -1.7442884 152 10
36.0 29.845779 6.1542206 153 10
34.7 33.790425 0.9095754 154 10
15.5 16.187347 -0.6873467 155 10
9.0 13.660144 -4.6601445 156 10
25.5 22.437895 3.0621053 157 10
12.0 13.932669 -1.9326686 158 10
27.9 20.828389 7.0716107 159 10
31.3 29.491329 1.8086710 160 10
17.0 18.890444 -1.8904444 161 10
18.1 19.424616 -1.3246164 162 10
15.0 15.668947 -0.6689469 163 10
26.0 27.526785 -1.5267847 164 10
19.0 20.587562 -1.5875617 165 10
27.2 31.144969 -3.9449686 166 10
20.0 19.161574 0.8384259 167 10
23.0 28.358612 -5.3586124 168 10
29.0 29.516167 -0.5161674 169 10
19.8 25.698870 -5.8988695 170 10
18.0 20.960275 -2.9602745 171 10
18.5 20.383049 -1.8830488 172 10
32.0 22.379860 9.6201397 173 10
21.6 19.839798 1.7602024 174 10
32.2 29.481610 2.7183904 175 10
36.0 31.424549 4.5754508 176 10
22.3 24.653504 -2.3535044 177 10
23.0 22.689484 0.3105160 178 10
21.0 25.335667 -4.3356670 179 10
14.0 14.882748 -0.8827481 180 10
19.0 21.703114 -2.7031141 181 10
34.1 33.210527 0.8894725 182 10
17.5 20.399456 -2.8994562 183 10
18.0 22.289167 -4.2891666 184 10
14.0 14.882748 -0.8827481 185 10
17.5 16.500025 0.9999749 186 10
19.4 16.454834 2.9451662 187 10
24.0 22.233911 1.7660889 188 10
24.0 20.292808 3.7071919 189 10
15.0 14.861914 0.1380861 190 10
36.1 32.855475 3.2445246 191 10
26.0 29.548565 -3.5485654 192 10
21.0 20.672111 0.3278892 193 10
16.0 20.986918 -4.9869179 194 10
31.5 32.227702 -0.7277023 195 10
17.0 20.407660 -3.4076600 196 10
16.0 21.734478 -5.7344782 197 10
27.0 23.989770 3.0102299 198 10
28.1 27.505193 0.5948066 199 10
16.0 20.986918 -4.9869179 200 10
22.0 29.182980 -7.1829805 201 10
16.0 14.992934 1.0070658 202 10
15.0 13.545962 1.4540381 203 10
20.0 21.692659 -1.6926594 204 10
24.0 22.680760 1.3192399 205 10
29.0 34.648584 -5.6485836 206 10
31.0 26.697332 4.3026676 207 10
20.0 24.674528 -4.6745280 208 10
25.0 29.559365 -4.5593647 209 10
43.1 32.590579 10.5094207 210 10
25.0 27.144327 -2.1443274 211 10
26.0 35.907151 -9.9071511 212 10
36.4 32.513162 3.8868379 213 10
23.0 25.316583 -2.3165826 214 10
19.2 15.585617 3.6143834 215 10
18.1 16.663504 1.4364958 216 10
33.7 29.465411 4.2345894 217 10
17.5 15.619345 1.8806545 218 10
28.0 27.904222 0.0957776 219 10
24.0 29.570164 -5.5701640 220 10
18.0 24.692809 -6.6928094 221 10
31.0 32.234533 -1.2345335 222 10
20.6 20.348593 0.2514067 223 10
26.4 24.616027 1.7839726 224 10
12.0 15.187671 -3.1876713 225 10
15.0 20.440475 -5.4404748 226 10
17.7 13.445775 4.2542249 227 10
19.1 24.059116 -4.9591162 228 10
22.0 21.671750 0.3282500 229 10
23.7 21.653977 2.0460230 230 10
20.5 20.926970 -0.4269701 231 10
28.0 23.409750 4.5902502 232 10
32.4 30.688480 1.7115201 233 10
13.0 13.752760 -0.7527599 234 10
14.0 13.700514 0.2994858 235 10
32.0 25.939976 6.0600241 236 10
13.0 11.264705 1.7352951 237 10
29.8 23.965192 5.8348083 238 10
43.4 32.552624 10.8473764 239 10
16.0 21.734478 -5.7344782 240 10
21.5 20.333827 1.1661734 241 10
30.0 32.248196 -2.2481959 242 10
37.7 33.979855 3.7201451 243 10
28.0 23.980992 4.0190079 244 10
14.5 14.589997 -0.0899973 245 10
18.0 22.289167 -4.2891666 246 10
27.5 22.650227 4.8497734 247 10
22.0 25.326125 -3.3261248 248 10
36.0 24.528277 11.4717235 249 10
33.0 29.878830 3.1211695 250 10
32.3 32.572502 -0.2725025 251 10
29.0 31.511840 -2.5118396 252 10
37.0 32.152559 4.8474412 253 10
34.0 24.546558 9.4534420 254 10
15.5 13.271760 2.2282403 255 10
22.0 22.047432 -0.0474323 256 10
44.6 32.394481 12.2055186 257 10
23.0 22.689484 0.3105160 258 10
15.5 15.659027 -0.1590266 259 10
16.2 17.754628 -1.5546285 260 10
15.0 30.890671 -15.8906711 261 10
18.0 21.713569 -3.7135688 262 10
25.0 22.672036 2.3279637 263 10
18.0 21.713569 -3.7135688 264 10
32.0 33.246014 -1.2460139 265 10
19.2 18.827078 0.3729224 266 10
20.0 21.361341 -1.3613410 267 10
20.0 21.692659 -1.6926594 268 10
26.0 32.663684 -6.6636841 269 10
37.0 23.332664 13.6673358 270 10
15.5 13.523708 1.9762923 271 10
15.0 12.129658 2.8703420 272 10
19.0 24.059994 -5.0599940 273 10
26.0 31.549250 -5.5492498 274 10
46.6 32.999299 13.6007010 275 10
14.0 16.583272 -2.5832724 276 10
25.0 23.435445 1.5645550 277 10
27.0 26.359371 0.6406286 278 10
29.9 33.281500 -3.3815003 279 10
29.5 31.117342 -1.6173423 280 10
14.0 14.471510 -0.4715103 281 10
36.0 34.781194 1.2188063 282 10
22.0 22.698208 -0.6982079 283 10
28.0 24.601402 3.3985977 284 10
32.0 31.474429 0.5255706 285 10
25.0 19.772227 5.2277728 286 10
34.3 28.240911 6.0590889 287 10
28.0 25.268872 2.7311282 288 10
12.0 13.656146 -1.6561457 289 10
19.0 22.279957 -3.2799573 290 10
19.0 20.176314 -1.1763142 291 10
14.0 14.882748 -0.8827481 292 10
14.0 14.882748 -0.8827481 293 10
34.1 32.192180 1.9078201 294 10
15.5 19.487741 -3.9877407 295 10
20.0 20.358438 -0.3584377 296 10
26.5 30.757039 -4.2570390 297 10
30.0 32.605791 -2.6057910 298 10
18.0 20.960275 -2.9602745 299 10
25.4 19.664630 5.7353704 300 10
26.0 28.327364 -2.3273643 301 10
23.9 22.234832 1.6651680 302 10
34.4 33.205458 1.1945420 303 10
14.0 14.882748 -0.8827481 304 10
16.0 14.841080 1.1589203 305 10
24.0 29.570164 -5.5701640 306 10
11.0 11.184232 -0.1842318 307 10
32.4 29.479450 2.9205503 308 10
28.8 19.696707 9.1032925 309 10
35.1 34.492059 0.6079411 310 10
13.0 13.752760 -0.7527599 311 10
17.6 25.720267 -8.1202674 312 10
18.0 21.713569 -3.7135688 313 10
13.0 14.903582 -1.9035823 314 10
20.0 22.715656 -2.7156556 315 10
12.0 13.932669 -1.9326686 316 10
26.0 35.907151 -9.9071511 317 10
29.8 34.173693 -4.3736925 318 10
38.1 34.389290 3.7107103 319 10
21.0 24.042438 -3.0424380 320 10
22.0 22.252330 -0.2523296 321 10
13.0 15.708628 -2.7086279 322 10
33.0 35.157206 -2.1572056 323 10
36.0 25.900425 10.0995749 324 10
21.1 22.706059 -1.6060593 325 10
34.2 31.446995 2.7530047 326 10
18.5 22.080900 -3.5808999 327 10
18.0 20.391253 -2.3912525 328 10
29.0 28.296116 0.7038838 329 10
14.0 14.471510 -0.4715103 330 10
27.0 34.769536 -7.7695356 331 10
17.5 16.500025 0.9999749 332 10
24.0 22.680760 1.3192399 333 10
33.5 31.455724 2.0442756 334 10
31.0 32.591318 -1.5913178 335 10
19.0 24.683669 -5.6836687 336 10
36.0 29.440572 6.5594279 337 10
17.0 21.724024 -4.7240235 338 10
20.3 21.206556 -0.9065556 339 10
23.5 20.301012 3.1989882 340 10
18.0 20.194020 -2.1940202 341 10
25.0 24.957484 0.0425158 342 10
26.0 31.930904 -5.9309040 343 10
21.5 19.841785 1.6582151 344 10
31.0 33.262912 -2.2629122 345 10
23.2 20.891001 2.3089986 346 10
11.0 11.262245 -0.2622454 347 10
38.0 32.490005 5.5099951 348 10
19.9 20.360079 -0.4600785 349 10
27.0 24.610543 2.3894570 350 10
20.2 25.694979 -5.4949790 351 10
13.0 13.530891 -0.5308907 352 10
40.8 33.097309 7.7026910 353 10
29.0 25.969639 3.0303610 354 10
32.1 31.473182 0.6268176 355 10
24.0 27.154541 -3.1545406 356 10
14.0 14.571206 -0.5712064 357 10
20.5 22.711294 -2.2112937 358 10
27.2 25.987437 1.2125631 359 10
18.0 24.068772 -6.0687720 360 10
27.0 24.610543 2.3894570 361 10
30.0 29.097924 0.9020756 362 10
23.9 24.016982 -0.1169819 363 10
23.0 18.717626 4.2823741 364 10
21.0 20.342030 0.6579697 365 10
35.7 27.427053 8.2729465 366 10
19.0 21.703114 -2.7031141 367 10
14.0 13.700514 0.2994858 368 10
19.0 24.683669 -5.6836687 369 10
32.0 34.370433 -2.3704330 370 10
30.0 27.485658 2.5143416 371 10
23.0 36.040900 -13.0408998 372 10
27.2 22.204442 4.9955585 373 10
33.5 23.932713 9.5672868 374 10
38.0 33.720342 4.2796576 375 10
13.0 16.607057 -3.6070573 376 10
13.0 13.752760 -0.7527599 377 10
29.0 26.339327 2.6606728 378 10
19.0 21.703114 -2.7031141 379 10
15.0 13.554222 1.4457785 380 10
44.3 32.438757 11.8612434 381 10
28.0 23.980992 4.0190079 382 10
18.0 14.799411 3.2005888 383 10
24.0 22.233911 1.7660889 384 10
26.0 23.998548 2.0014519 385 10
25.0 22.672036 2.3279637 386 10
21.0 25.687198 -4.6871979 387 10
14.0 14.571206 -0.5712064 388 10
28.0 29.526967 -1.5269667 389 10
22.0 22.939785 -0.9397846 390 10
30.0 31.499370 -1.4993696 391 10

Leave-one-out Cross-Validation

Get MSE for all levels of flexibility (power)

error_obs_tab |>
    group_by(pow) |>
    summarise(
        mse = mean(`.resid`^2, na.rm = T)
    )|>
    ggplot(aes(pow,mse),colour = "blue")+
    geom_point()+
    geom_line()+
    labs(
    x = "Flexibility",
    y = "MSE"
    )+
    theme_minimal()

Leave-one-out Cross-Validation

Leave-one-out Cross-Validation

LooCV is less biased than the Validation set approach as it uses almost all the data to train the model.

Test error rate estimates of LooCV are not as spread out as provided by validation set approach as it does not get affected by randomness of splits.

Though it is expensive to use computationally, especially for data that is large.

k-Fold Cross-Validation

k-Fold CV addresses the computational issue with LooCV, LooCV is a soecial case of k-fold

The data is split in k parts, all parts should be nearly equal.

The first part is used as a Validation set, model is fit on the remaining k-1 parts.

This is repeated till each of the k parts have been used as the validation set.

Test error rate is estimated as the average of all k validation set error rates.

k-Fold Cross-Validation

Begin by creating the splits

rsample::vfold_cv(Auto,v = 10) -> Auto_kfold_splits

Auto_kfold_splits
#  10-fold cross-validation 
# A tibble: 10 × 2
   splits           id    
   <list>           <chr> 
 1 <split [352/40]> Fold01
 2 <split [352/40]> Fold02
 3 <split [353/39]> Fold03
 4 <split [353/39]> Fold04
 5 <split [353/39]> Fold05
 6 <split [353/39]> Fold06
 7 <split [353/39]> Fold07
 8 <split [353/39]> Fold08
 9 <split [353/39]> Fold09
10 <split [353/39]> Fold10

k-Fold Cross-Validation

Function to get \(MSE_{k}\)

get_k_mse <- function(k,pow){

    Auto_ktrain <- rsample::training(Auto_kfold_splits$splits[[k]])

    Auto_ktest <- rsample::testing(Auto_kfold_splits$splits[[k]])

    mod<- lm(mpg ~ poly(horsepower,pow), Auto_ktrain)

    tibble(
        k = k,
        pow = pow,
        k_mse = broom::augment(
            mod,
            newdata = Auto_ktest
            ) |>
            summarise(
                mse = mean(`.resid`^2, na.rm=T)
            ) |>
                pull(mse)

    )

}

k-Fold Cross-Validation

Execute function to get \(CV_k\) at different powers

pmap(
    list(
        k = rep(c(1:10),10),
        pow = rep(c(1:10),10) |> sort()
    ),

    get_k_mse
) |>
    list_rbind() |>
    group_by(pow) |>
    summarise(
        mse = mean(k_mse, na.rm = T)
    ) |>
    ggplot(aes(pow,mse),colour = "blue")+
    geom_point()+
    geom_line()+
    labs(
    x = "Flexibility",
    y = "MSE"
    )+
    theme_minimal()

k-Fold Cross-Validation

k-Fold Cross-Validation

Create 10 reps of the k-fold split

rsample::vfold_cv(Auto,v = 10, repeats = 100)-> Auto_kfold_splits_100reps

Auto_kfold_splits_100reps
#  10-fold cross-validation repeated 100 times 
# A tibble: 1,000 × 3
   splits           id        id2   
   <list>           <chr>     <chr> 
 1 <split [352/40]> Repeat001 Fold01
 2 <split [352/40]> Repeat001 Fold02
 3 <split [353/39]> Repeat001 Fold03
 4 <split [353/39]> Repeat001 Fold04
 5 <split [353/39]> Repeat001 Fold05
 6 <split [353/39]> Repeat001 Fold06
 7 <split [353/39]> Repeat001 Fold07
 8 <split [353/39]> Repeat001 Fold08
 9 <split [353/39]> Repeat001 Fold09
10 <split [353/39]> Repeat001 Fold10
# ℹ 990 more rows

k-Fold Cross-Validation

Function to Repeat the process for every power(1:10) 10 times

get_k_repeats_mse <- function(itr, pow){
    
    Auto_ktrain <- rsample::training(Auto_kfold_splits_100reps$splits[[itr]])

    Auto_ktest <- rsample::testing(Auto_kfold_splits_100reps$splits[[itr]])

    mod <- lm(mpg ~ poly(horsepower,pow), Auto_ktrain)

    broom::augment(mod, newdata = Auto_ktest) |>
        select(`.resid`) |>
        mutate(
            pow = pow,
            rep = Auto_kfold_splits_100reps$id[itr],
            fold = Auto_kfold_splits_100reps$id2[itr]
        )  

}

k-Fold Cross-Validation

pmap(
    list(
        itr = c(1:1000),
        pow = rep(c(1:10),100) |> sort()
    ),
    get_k_repeats_mse
) |>
    list_rbind() |> 
    group_by(rep,fold,pow)|>
    summarise(
        mse = mean(`.resid`^2, na.rm = T)
    ) |> 
        ungroup()|>
        group_by(rep,pow)|>
        summarise(
        mse = mean(mse, na.rm = T)
        ) |> 
    ungroup()|> 
    ggplot(aes(pow,mse,colour = rep))+
    geom_point()+
    labs(
    x = "Flexibility",
    y = "MSE"
    )+
    theme_minimal() +
    theme(
        legend.position = "none"
    )

k-Fold Cross-Validation

k-Fold Cross-Validation

k-Fold CV is the practical choice for most methods.

It balances the bias problem of validation set approach and the computational constraint of the LooCV

Due to the Bias-Variance trade-off it offers better estimation of test error rate compared to LooCV

Appling CV using {boot}

glm.fit <- glm(mpg ~ poly(horsepower,2), data = Auto)

boot::cv.glm(Auto, glm.fit, K = 10)$delta
[1] 19.31836 19.30109